Calculating the spectral factorization and outer functions by sampling-based approximations-Fundamental limitations

JOURNAL OF APPROXIMATION THEORY

卷:257

Article

Boche, Holger^1,2  Pohl, Volker¹ 

[1] Tech Univ Munich, Chair Theoret Informat Technol, Arcisstr 21, D-80333 Munich, Germany

[2] Munich Ctr Quantum Sci & Technol MCQST, Schellingstr 4, D-80799 Munich, Germany

关键词: Approximation;    Sampling;    Spectral factorization;    Outer functions;    Finite energy;    Uniform recovery;   

DOI  :  10.1016/j.jat.2020.105450

来源: Elsevier

【 摘 要 】

This paper considers the problem of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of these spectral densities. Although there exists a closed form expression for the spectral factor, this formula shows a very complicated behavior because of the non-linear dependency of the spectral factor from spectral density and because of a singular integral in this expression. Therefore approximation methods are usually applied to calculate the spectral factor. It is shown that there exists no sampling-based method which depends continuously on the samples and which is able to approximate the spectral factor for all densities in this set. Instead, to any sampling-based approximation method there exists a large set of spectral densities so that the approximation method does not converge to the spectral factor for every spectral density in this set as the number of available sampling points is increased. The paper will also show that the same results hold for sampling-based algorithms for the calculation of the outer function in the theory of Hardy spaces. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free